Astronomy & Astrophysics manuscript no. recovery_paper December 12, 2016
c
ESO 2016
Evolution of the real-space correlation function from next generation cluster surveys Recovering the real-space correlation function from photometric redshifts Srivatsan Sridhar1? , Sophie Maurogordato1 , Christophe Benoist1 , Alberto Cappi1, 2 , and Federico Marulli2, 3, 4 1 2
arXiv:1612.02821v1 [astro-ph.CO] 8 Dec 2016
3 4
Université Côte d’Azur, OCA, CNRS, Lagrange, UMR 7293, CS 34229, 06304, Nice Cedex 4, France. e-mail:
[email protected] INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127, Bologna, Italy. Dipartimento di Fisica e Astronomia - Università di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy INFN - Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna,Italy
Received xxx; Accepted 01/12/2016 ABSTRACT Aims. In this paper, we investigate to which accuracy it is possible to recover the real-space two-point correlation function of galaxy
clusters from cluster catalogues based on photometric redshifts, and test our ability to measure the redshift and mass evolution of the correlation length r0 and of the bias parameter b(M,z) as a function of the redshift uncertainty. Methods. We calculate the correlation function for cluster sub-samples covering various mass and redshift bins selected from a 500 deg2 light-cone catalogue. To simulate the distribution of clusters in photometric redshift space, we assign to each cluster a σz redshift randomly extracted from a Gaussian distribution. The dispersion is varied in the range σ(z=0) = 1+z = 0.005, 0.010, 0.030 and c 0.050. The correlation function in real-space is computed through estimation and deprojection of w p (r p ). Four mass ranges (from Mhalo > 2 × 1013 h−1 M to Mhalo > 2 × 1014 h−1 M ) and six redshift slices covering the redshift range [0,2] are investigated, using cosmological redshifts and photometric redshift configurations. Results. From the analysis in cosmological redshifts we find a clear increase of the correlation amplitude as a function of redshift and mass. The evolution of the derived bias parameter b(M,z) is in agreement with theoretical expectations. From our pilot sample limited to Mhalo > 5 × 1013 h−1 M (0.4 < z < 0.7), we find that the real-space correlation function can be recovered by deprojection of w p (r p ) within an accuracy of 5% for σz = 0.001 × (1 + zc ) and within 10% for σz = 0.03 × (1 + zc ). The evolution of the correlation in redshift and mass is clearly detected for all σz tested, but requires a large binning in redshift to be detected significantly between individual redshift slices when increasing σz . The best-fit parameters (r0 and γ) as well as the bias obtained from the deprojection method for all σz are within the 1σ uncertainty of the zc sample. Key words. galaxies: clusters: general – (cosmology:) large-scale structure of Universe – techniques: photometric – methods: statistical
1. Introduction One of the major challenges in modern cosmology is to explain the observed acceleration of the cosmic expansion, determining if it is due to a positive cosmological constant, a time–varying dark energy component or a modified theory of gravity. Major galaxy surveys are currently ongoing or in preparation in order to address this fundamental question through the analysis of various complementary cosmological probes with different systematics, as for instance weak lensing, galaxy clustering (baryon acoustic oscillations, redshift-space distortions) and galaxy clusters. In fact galaxy cluster counts as a function of redshift and mass are sensitive to dark energy through their dependence on the volume element and on the structure growth rate. One intrinsic difficulty in constraining the cosmological models with galaxy cluster counts comes from uncertainties in cluster mass estimates and on the difficulty to calibrate related mass proxies. One can overcome this difficulty adding the information related to the clustering properties of clusters, due to the fact that their power spectrum amplitude depends mainly on the halo mass. When combining the redshift-averaged cluster power spectrum and the evolution of the number counts in a given survey, the constraints on the dark ?
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energy equation of state are dramatically improved (Majumdar & Mohr 2004). Recent cosmological forecasting based on galaxy clusters confirms that the figure of merit significantly increases when adding cluster clustering information (Sartoris et al. 2016). Cluster clustering can be measured through the two–point correlation function, the Fourier transform of the power spectrum, which is one of the most successful statistics for analysing clustering processes (Totsuji & Kihara 1969; Peebles 1980). In cosmology, it is a standard tool to test models of structure formation and evolution. The cluster correlation function is much higher than that of galaxies, as first shown by Bahcall & Soneira (1983) and Klypin & Kopylov (1983). This is a consequence of the fact that more massive haloes correspond to higher and rarer density fluctuations, which have a higher correlation amplitude (Kaiser 1984). Galaxy clusters are associated to the most massive virialised dark matter haloes, and as a consequence their correlation function is strongly amplified. The evolution of the cluster halo mass, bias and clustering has been addressed analytically (Mo & White 1996; Moscardini et al. 2000; Sheth et al. 2001), and also numerically (Governato et al. 1999; Angulo et al. 2005; Estrada et al. 2009). The increase of the correlation length with cluster mass and redshift has been used to constrain the cosmoArticle number, page 1 of 21
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Random distribution Data distribution
30000 25000
N(z)
20000 15000 10000 5000 0 0.0
0.5
1.0
1.5
z
2.0
2.5
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information to reproduce the photometric uncertainty on redshift expected in future cluster experiments. The paper is organised as follows. In Section 2 we describe the simulation we work with. In Section 3 we investigate the two-point correlation function evolution with redshift and mass without any error on the redshift to check consistency with theory. The bias is calculated for different mass cut samples along with the evaluation of clustering strength with mass at different redshift and compared with the theoretical expectation of Tinker et al. (2010). We also calculate the mean intercluster comoving separation (d) and compare it with r0 and perform an analytic fit to this r0 vs d relation. Section 4 presents the deprojection method we use to recover the real-space correlation function from mock photometric catalogues generated using a Gaussian approximation technique. The results obtained from the deprojection method along with the redshift evolution of the samples with redshift uncertainty are presented. In Section 5, the overall results obtained from our study are summarized and discussed.
Fig. 1: Redshift distribution of the entire catalogue with 0.0 < zc < 3.0. The data distribution is shown as the histogram along 2. Simulations with the blue line specifying the distribution of the random catalogue we use for calculating the two-point correlation function. We use a public light-cone catalogue constructed using a semianalytic model of galaxy formation (Merson et al. 2013) onto the N-body dark matter halo merger trees of the Millennium Simulation, based on a ΛCDM cosmological model with the logical model and the bias (Colberg et al. 2000; Bahcall et al. following parameters: Ω M , ΩΛ , Ωb , h = 0.25, 0.75, 0.045, 0.73 2004; Younger et al. 2005) (Springel et al. 2005), corresponding to the first year results from The first large local surveys such as the SDSS (Eisenstein et al. the Wilkinson Microwave Anisotropy Probe (Spergel et al. 2007). 2011) have led to a significant progress in this field. Clustering The Millennium simulation was carried out using a modified properties of cluster catalogues derived from SDSS were done version of the GADGET2 code (Springel 2005). Haloes in the by Estrada et al. (2009), Hütsi (2010) and Hong et al. (2012). simulation were resolved with a minimum of 20 particles, with a More recently, Veropalumbo et al. (2014) have shown the first resolution of Mhalo = 1.72×1010 h−1 M (M represent the mass unambiguous detection of the BAO peak in a spectroscopic sam- of the Sun). The groups of dark matter particles in each snapshot ple of 25000 clusters selected from the SDSS, and measured were identified through a Friends-Of-Friends algorithm (FOF) the peak location at 104 ± 7 h−1 Mpc . Large surveys at higher following the method introduced by Davis et al. (1985). redshifts which are ongoing (DES, BOSS, KIDS, Pan-STARRS) However, the algorithm was improved with respect to the or in preparation (eROSITA, LSST, Euclid) open a new window original FOF, to avoid those cases where the FOF algorithm for the analysis of cluster clustering. The wide areas covered merge groups connected for example by a bridge, while they will give access to unprecedented statistics (∼100,000 clusters should be considered instead as separated haloes (Merson et al. expected with Dark Energy Survey, eROSITA and Euclid survey) 2013). that will allow us to cover the high mass, high redshift tail of The linking length parameter for the initial FOF haloes is the mass distribution, to control cosmic variance, and to map the b = 0.2. Notice however that haloes were identified with a method large scales at which the BAO signature is expected (∼ 100Mpc). different from the standard FOF. The FOF algorithm was initially However, to use clusters as cosmological probes, among the applied to find the haloes, but then their substructures were idenseveral difficulties to be overcome is the impact of photometric tified using the so-called SUBFIND algorithm: depending on the redshift errors. While some of these surveys have (in general par- evolution of the substructures and their merging, a new final halo tially) a spectroscopic follow up, many forthcoming large galaxy catalogue was built. The details of this method are described by surveys will have only the photometric information in multiple Jiang et al. (2014). bands, so that their cluster catalogues will be built on the basis A comparison between the masses obtained with this imof state-of-the-art photometric redshifts. Using those instead of proved D-TREES algorithm, Mhalo , and the classical MFOF , and real redshifts will cause a positional uncertainty along the line their relation with M200 , was done by Jiang et al. (2014), where of sight inducing a damping of clustering at small scales and a it is shown that at redshift z = 0 on average, Mhalo overestimates smearing of the acoustic peak (Estrada et al. 2009). It is there- M200 , but by a lower factor with respect to MFOF : they found fore of major interest to check the impact of this effect on the that only 5% of haloes have Mhalo /M200 > 1.5. However, when recovery of the real-space correlation function. Our objective is comparing the halo mass function of the simulation with that to optimize the analysis of cluster clustering from forthcoming expected from the Tinker et al. (2010) approximation, it appears cluster catalogues that will be issued from the ongoing/future that there is a dependence on redshift, and beyond z ≈ 0.3 the large multiband photometric surveys. Here we focus on the deter- Mhalo /M200 ratio becomes less than 1 (Mauro Roncarelli, private mination of the two-point correlation function, and the aim of our communication). This has to be taken into account in further paper is i) to determine the clustering properties of galaxy clus- analysis using the masses. ters from state-of-the-art simulations where galaxy properties are Galaxies were introduced in the light-cone using the Lagos 12 derived from semi-analytical modelling (Merson et al. 2013), and GALFORM model (Lagos et al. 2012). The GALFORM model ii) to test how much the clustering properties evidenced on ideal populates dark matter haloes with galaxies using a set of difmock catalogues can be recovered when degrading the redshift ferential equations to determine how the baryonic components Article number, page 2 of 21
Srivatsan Sridhar et al.: Evolution of the real-space correlation function from next generation cluster surveys
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10
. 5 × 1013 h−1 M , Mhalo > 1 × 1014 h−1 M and Mhalo > 2 × 1014 h−1 M . The analysis is performed in the same redshift slices previously defined. The correlation function is fitted with a power-law as it can be seen from Figure 2b, both with a free slope and with a fixed slope γ = 2.0. The mass range, the values of the best-fit parameters, the number of clusters, and the bias for each sub-sample are given in the four panels of Table 1. Each panel corresponds to a different selection in mass. In both cases, the correlation length r0 increases with the limiting mass at any redshift and increases with redshift at any limiting mass, as shown in Figure 4. The higher the mass threshold, the larger is the increase of r0 with redshift. For example, the ratio of the correlation lengths for the [1.3-1.6] and the [0.1-0.4] redshift slices is 1.25 with Mhalo > 2 × 1013 h−1 M , while it reaches 1.8 with Mhalo > 1 × 1014 h−1 M . For the largest limiting mass (Mhalo > 2 × 1014 h−1 M ), the number of clusters becomes small at high z and the analysis must be limited to z = 1. We can again compare our results with the analysis of Younger et al. (2005). who used a Tree Particle Mesh (TPM) code (Bode & Ostriker 2003) to evolve N = 12603 particles in a box of 1500 h−1 Mpc, reaching a redshift z ≈ 3.0. We find a good agreement (see their Figure 5) for the common masses and
∆ρ ρ
! =b× light
∆ρ ρ
! (5a) mass
where b is the bias factor and ρ is the density. The higher the halo mass, the higher the bias. The amplitude of the halo correlation function is amplified by a b2 factor with respect to the matter correlation function: ξ(r)light = b2 × ξ(r)mass
(5b)
The amplitude of the matter correlation function increases with time and decreases with redshift, but the halo bias decreases with time and increases with redshift. As a result, the cluster correlation amplitude increases with redshift, as it is clear in Figure 3. We estimate the cluster bias through Equation 5b. The power spectrum of the dark matter distribution is calculated with the cosmological parameters of the light-cone we use and we obtain its Fourier transform ξ(r). We use the function xi_DM from the class Cosmology from CosmoBolognaLib. The comparison is not straightforward because, as we have previously noticed, in the simulation halo masses are not M200 , but the so–called Dhalos masses Mhalo . The evolution of bias with redshift for the 4 sub-samples with different limiting masses is shown in Figure 5 along with the theoretical predictions of Tinker et al. (2010) for the same limiting masses used. The values of the bias for the different sub-samples are provided in Table 1. It can be seen that at fixed redshifts more massive clusters have a higher bias; at fixed mass threshold the bias increases with redshift, and evolves faster at higher redshifts. It can also be seen clearly that the bias obtained for the haloes from the simulations are in good agreement with the predictions by Tinker et al. (2010). At high redshifts (z > 0.8) the bias recovered from the simulations seems to slightly diverge from the theoretical predictions, especially for the Mhalo > 1 × 1014 h−1 M sample, but this can be explained by the dependence on redshift for the Mhalo /M200 ratio which becomes smaller than 1 at these redshifts as previously mentioned. The discrepancy is not significant as our bias measurements are well within 1σ precision from the theoretical expectations. 3.5. The r0 - d relation
The dependence of the bias on the cluster mass is based on theory. A complementary and empirical characterization of the cluster correlation function is the dependence of the correlation length r0 as a function of the mean cluster comoving separation d (Bahcall Article number, page 5 of 21
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Table 1: The best-fit values of the parameters of the real-space correlation function ξ(r) for the light-cone at different (1) mass thresholds and (2) redshift ranges. For each sample we quote (3) the correlation length r0 , (4) slope γ, (5) correlation length r0 at fixed slope γ = 2.0, (6) number of clusters Nclusters and (7) the bias b obtained. Mass (h−1 M )
Mhalo > 2 × 1013
Mhalo > 5 × 1013
Mhalo > 1 × 1014
Mhalo > 2 × 1014
z
r0 (h−1 Mpc)
γ
r0 (γ = 2.0) (h−1 Mpc)
Nclusters
bias
0.1 < zc < 0.4
9.89±0.20
1.76±0.05
9.53±0.29
10492
1.81±0.03
0.4 < zc < 0.7
10.22±0.14
1.84±0.04
10.01±0.17
27224
2.00±0.03
0.7 < zc < 1.0
11.10±0.15
1.87±0.04
10.85±0.17
35133
2.52±0.02
1.0 < zc < 1.3
11.62±0.23
1.98±0.05
11.58±0.19
31815
3.01±0.06
1.3 < zc < 1.6
12.41±0.42
2.13±0.09
12.49±0.52
22978
3.37±0.19
1.6 < zc < 2.1
14.78±0.21
2.06±0.05
14.78±0.22
18931
4.65±0.23
0.1 < zc < 0.4
12.22±0.26
1.90±0.05
11.97±0.25
3210
2.21±0.05
0.4 < zc < 0.7
13.20±0.23
1.98±0.05
13.16±0.17
7301
2.62±0.13
0.7 < zc < 1.0
14.86±0.33
1.97±0.05
14.52±0.28
8128
3.38±0.22
1.0 < zc < 1.3
17.00±0.48
2.00±0.07
17.00±0.38
5963
4.38±0.19
1.3 < zc < 1.6
18.26±0.62
2.15±0.06
19.73±0.43
3365
5.29±0.31
1.6 < zc < 2.1
19.18±1.41
2.23±0.21
20.05±1.13
2258
6.21±0.62
0.1 < zc < 0.4
14.60±0.35
1.93±0.06
14.33±0.24
1119
2.67±0.19
0.4 < zc < 0.7
17.26±0.96
1.90±0.08
16.35±0.42
2228
3.45±0.23
0.7 < zc < 1.0
18.93±1.18
2.08±0.12
19.55±0.75
2072
4.64±0.37
1.0 < zc < 1.3
22.36±1.90
2.11±0.17
23.33±1.30
1221
6.15±0.82
1.3 < zc < 1.6
26.09±4.10
2.28±0.30
28.96±3.17
590
7.64±2.50
0.1 < zc < 0.4
19.98±1.92
1.95±0.22
19.73±1.22
322
3.98±0.38
0.4 < zc < 0.7
22.23±1.54
2.16±0.18
22.27±1.17
538
4.57±0.45
0.7 < zc < 1.0
24.65±1.89
2.19±0.29
25.28±1.68
407
6.01±1.63
& Soneira 1983; Croft et al. 1997; Governato et al. 1999; Bahcall et al. 2003), where: d=
q 3
1 ρ
(6)
and ρ is the mean number density of the cluster catalogue on a given mass threshold. According to the theory, more massive clusters have a higher bias, therefore a higher r0 ; as they are also more rare, they have also a larger mean separation: therefore it is expected that r0 increases with d, i.e. r0 = α d β . This relation has been investigated both in observational data (Bahcall & West 1992; Estrada et al. 2009) and in numerical simulations (Bahcall et al. 2003; Younger et al. 2005). Younger et al. (2005) gave an analytic approximation in the ΛCDM case in the redshift range z = 0-0.3 for 20 ≤ d ≤ 60 h−1 Mpc, with α = 1.7 and β = 0.6. We determine the r0 dependence with d for the various subsamples previously defined. The results obtained for a free γ along with the best fit obtained for both free and fixed γ = 2 are shown in Figure 6. The best-fit parameters for the r0 - d relation in the redshift range 0 ≤ z ≤ 2.1 and for the cluster mean separation range 20 ≤ d ≤ 140h−1 Mpc are, α = 1.77 ± 0.08 and β = 0.58 ± 0.01. Article number, page 6 of 21
The r0 - d relation which appears to be scale–invariant with redshift, is consistent with what was found by Younger et al. (2005) and is also consistent with the theoretical predictions of Estrada et al. (2009) (see their figure 7). The scale invariance of the r0 − d relation up to a redshift z ≈ 2.0 implies that the increase of the cluster correlation strength with redshift is matched by the increase of the mean cluster separation d. It suggests that the cluster mass hierarchy does not evolve significantly in the tested redshift range: for example, the most massive clusters at an earlier epoch will still be among the most massive at the current epoch.
4. Estimating the correlation function with photometric redshifts Large redshift surveys such as the Sloan Digital Sky Survey (Eisenstein et al. 2011), VVDS (Le Fèvre et al. 2005), VIPERS (Garilli et al. 2014; Guzzo et al. 2014) have revolutionized our tridimensional vision of the Universe. However, as spectroscopic follow up is a very time consuming task, priority has been given either to the sky coverage or to the depth of the survey. An alternative way of recovering the redshift information is to derive it from imaging in multiple bands when available, using the technique of photometry (Ilbert et al. 2006, 2009) While the accuracy of spectroscopic redshifts cannot be reached, this method can
Srivatsan Sridhar et al.: Evolution of the real-space correlation function from next generation cluster surveys
Table 3: The best-fit parameters obtained for the real-space correlation function ξ(r) of the original sample and the recovered deprojected correlation function ξdep (r) for the mock photometric redshift samples. We quote the (1) redshift uncertainty, (2) the correlation length r0 , (3) slope γ. The mass cut used is Mhalo > 5 × 1013 h−1 M and the fit range is fixed at 5-50 Mpc. The fits have been performed both with fixed (γ = 2.0) and free slope.
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